Our goal in this section is to compute a flow function \(\psi \)  homologous to a given flow function \(\phi \)  such that
\(\psi \)  represents a \(T\) -linkage.
To control the number of edges with \(\psi (\cdot )\ne \epsilon \)  incident to a common vertex,
Schrijver [1]} introduced a more general problem setting: the homology feasibility problem.
In this problem, we are given a flow function \(\phi \) , and a candidate function \(\Gamma : {\Pi } \rightarrow 2^{\Sigma ^*}\) ,
where \(\Pi \)  is a set of face-edge paths.
Here, a face-edge path in \(G^\textsf {p}\)  is an alternating sequence of edges and faces such that
two consecutive edge and face are adjacent in \(G^\textsf {p}\) .
Moreover, \(\Gamma (\pi )\)  must be hereditary, that is,
for every string \(x\in \Gamma (\pi )\) , all its prefixes and \(x^{-1}\)  also belong to \(\Gamma (\pi )\) .
In the following, for a flow function \(\psi \) , we let
\(\psi (\pi )\)  denote the product of \(\psi (e)^{\mu (e)}\) 's for all edges \(e\)  in \(\pi \) ,
where the sign \(\mu (e)\)  is positive if \(L_e\)  is the face lying previous to \(e\)  in \(\pi \) , and negative, otherwise.
